Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Find the domain of a function $f(x,y)=\log_xy$. Does $f\in C^1$? (I don't know if this symbol is common - it means that $f$ is at least once differentiable, and its first derivative is continuous).

from basic facts the domain is: $D=\mathbb{R}^2 \setminus \left\{ (x,y) : x=1 \vee y\le 0 \vee x\le 0 \right\}$

partial derivatives (maybe it will useful):

$\frac{\partial}{\partial x}f(x,y)=-\frac{\ln y}{x\ln^2 x}$

$\frac{\partial}{\partial y}f(x,y)=\frac{1}{y\ln x}$

but I don't understand what is necessary for $f\in C^1$. Do I just need to check if partial derivatives are both continuous for all $x,y\in D$?

I would be very grateful for help.

share|cite|improve this question
up vote 0 down vote accepted

The statement $f \in C^1$ means that $f$ is continuously differentiable.

There is a theorem that says that if each of the partials exists and is continuous at $x$, then the function is continuously differentiable at $x$.

In your case, the partials exist and are continuous on $D$, hence $f \in C^1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.